I am not a mathematician, my only higher level mathematics course was, what would now be called, a service course. I took this course over 40 years ago and the approach was very practical. So as I work through the book on measure theory I am having to look up definitions which would probably be obvious to a newer mathematics graduate.
This page records those definitions.
The Axiom of Choice – if a set is partitioned into a number of non-empty disjoint subsets, , the axiom of choice says that we can always obtain a set, , by choosing one element from each of the . This seems self evident when the number of subsets is finite; the axiom of choice extends this to the case where the number of subsets is infinite.
Countable – a set, , is countable if one can define a bijective mapping from to a subset of the set of natural numbers, . If the mapping is from to then is said to be countably infinite.
Infimum – given a set , the infimum, , is the greatest lower bound of . For example, if , there is no minimum value, but is defined and equals . Note that ,
Supremum – given a set , the supremum, , is the least upper bound of . For example, if , there is no maximum value, however is defined and equals . Note that ,
Uncountable – a set is uncountable if, and only if, it is not finite and not countable. The set of real numbers is uncountable.